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Questions  >>  CBSE XI  >>  Math  >>  Trigonometric Functions
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Q)

If $(1+\tan A)(1+\tan B)=2$ find all values of $A+B$

$\begin{array}{1 1}(A)\;n\pi+\large\frac{\pi}{4}&(B)\;n\pi+ \large\frac{\pi}{6}\\(C)\;n\pi+ \large\frac{\pi}{8}&(D)\;2n\pi+\large\frac{3\pi}{4}\end{array} $

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A)
Toolbox:
  • $\tan A+B=\large\frac{\tan A+\tan B}{1-\tan A\tan B}$
  • $\tan \theta=\tan \alpha\Rightarrow \theta=n\pi+\alpha$
$(1+\tan A)(1+\tan B)=2$
$1+\tan A+\tan B+\tan A\tan B=2$
$\tan A+\tan B=1-\tan A.\tan B$
$\large\frac{\tan A+\tan B}{1-\tan A.\tan B}$$=1$
$\tan(A+B)=1=\tan\large\frac{\pi}{4}$
$\Rightarrow A+B=n\pi+\large\frac{\pi}{4}$ where $n=0,\pm 1,\pm 2$
Hence (A) is the correct answer.
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